A Finite Volume Scheme for Transient Nonlocal Conductive-Radiative Heat Transfer, Part 2: Convergence to the Weak Solution

Peter Philip, IMA

Convergence is proved for a finite volume scheme for transient nonlinear heat transport equations coupled by nonlocal interface conditions. The interface conditions model diffuse-gray radiation between the surfaces of (both open and closed) cavities. The model is considered in three space dimensions. The special difficulties of the problem lie in the radiative nonlocal coupling between surfaces and in the allowed nonlinear dependence of internal energy and emissivities on the solution (i.e. temperature). Moreover, at material interfaces, the internal energy and the (otherwise constant) diffusion coefficient can be discontinuous. For each time and space discretization, the finite volume scheme gives rise to a piecewise constant interpolation. It is shown that, if finenesses of the time and space discretization tend to 0, then a subsequence of the corresponding interpolations converges to a weak solution of the continuous problem. A discrete maximum pinciple allows to prove a discrete H1-estimate as well as estimates of time and space translates. Convergence is then based on the Kolmogorov compactness theorem.